Question 169: Thermodynamics & KTG

Question

Energy of 10 non rigid diatomic molecules at temperature $\mathrm{T}$ is :

Accepted Answer

The energy of a diatomic molecule depends on the degrees of freedom it has. For a non-rigid diatomic molecule, there are more degrees of freedom compared to a rigid diatomic molecule. Specifically, a non-rigid diatomic molecule has translational, rotational, and vibrational degrees of freedom. The translational and rotational degrees of freedom are the same for both rigid and non-rigid diatomic molecules, which include:
3 translational degrees of freedom,
2 rotational degrees of freedom (since linear molecules do not have the third rotational freedom because it requires rotating around the bond axis, which doesn't change the energy for linear molecules).
Additionally, non-rigid diatomic molecules have vibrational degrees of freedom. For a simple diatomic molecule, there is 1 vibrational degree of freedom (since vibration along the bond axis is possible). However, considering the energy distribution across these vibrations requires accounting for both the potential and kinetic energy associated with vibrations, effectively doubling the vibrational degrees of freedom for energy calculations to 2 (one for kinetic and one for potential energy).
Thus, the total degrees of freedom for a non-rigid diatomic molecule are:
3 (translational) + 2 (rotational) + 2 (vibrational) = 7 degrees of freedom.
The formula for the average energy of a molecule in terms of degrees of freedom at temperature T is given by:
$ E = \frac{f}{2}k_\mathrm{B}T $
where $f$ is the total number of degrees of freedom, $k_\mathrm{B}$ is the Boltzmann constant, and $T$ is the temperature.
Plugging in the values for a non-rigid diatomic molecule:
$ E = \frac{7}{2}k_\mathrm{B}T $
Therefore, the total energy for 10 such molecules would be:
$ 10 	imes \frac{7}{2}k_\mathrm{B}T = 35k_\mathrm{B}T $
So, the correct answer is:
Option D: 35 $k_\mathrm{B}T$.

Answer

35 RT

Answer

$\frac{7}{2}$ RT

Answer

70 K
B
T

Answer

35 K
B
T

Thermodynamics & KTG

Energy of 10 non rigid diatomic molecules at temperature \mathrm{T} is :

Choose the correct answer: